The generalized p-Wasserstein distance, for $p\geq 1$, is given by $$d_W(Q_1,Q_2):=inf \left\{\int_{\Xi_2}||\xi_1-\xi_2||^p \Pi(d\xi_1,d\xi_2)\right\}$$ where $\Pi$ is the joint distribution of $\xi_1$ and $\xi_2$ with marginals $Q_1$ and $Q_2$, respectively. I am using this in relation to distributional robust optimization.
My question is; are there any intuition on which p-norm to use, and how it impacts the Wasserstein distance. For instance, in relation to distributional robust optimization, the 1-norm and $\infty$-norm leads to linear programs for some specific loss functions of practical interest. This is ofc. a nice feature, but would, e.g. the 2-norm lead to "better results" for multivariate distributions.
I am wondering about this, because I am using data over space and time, where the data is highly correlated over the space dimension. I was thus wondering if it would be more suitible to use the 2-norm, instead of the $\infty$-norm (which I am using now).
